INTERNATIONAL GONGRESS $44Q996\; 639\; 977\; 612T954\; 581Q953\; 580\; 938\; 580Q909\; 582\; 884\; 587L869\; 591L870\; 587Q870\; 583\; 849\; 557T796\; 491T748\; 422Q729\; 391\; 692\; 313T620\; 188Q555\; 105\; 454\; 34T253\; -37Q214\; -37\; 181\; -30T120\; -2T92\; 53Q92\; 89\; 119\; 123T184\; 158Q205\; 158\; 215\; 146T225\; 119Q225\; 102\; 203\; 89T161\; 75Q153\; 75\; 145\; 78T135\; 81Q130\; 81\; 130\; 62Q130\; 39\; 153\; 24T204\; 5T267\; 0Q311\; 0\; 358\; 29T454\; 117T539\; 226T629\; 358T710\; 476Q726\; 496\; 744\; 516T778\; 551T807\; 577T828\; 595L836\; 601L785\; 623Q743\; 642\; 713\; 651T668\; 661T626\; 663Q564\; 663\; 509\; 644T418\; 596T356\; 535T317\; 475T305\; 431Q305\; 416\; 312\; 408Q323\; 388\; 369\; 388Q429\; 388\; 465\; 411T530\; 480Q557\; 526\; 557\; 565Q557\; 573\; 556\; 579T555\; 587T555\; 590Q555\; 591\; 568\; 600T584\; 611Q588\; 612\; 600\; 603Q622\; 581\; 622\; 549Q622\; 516\; 600\; 475T536\; 405Q454\; 350\; 354\; 350Z">$ {\mathcal{T}}_c^b$$;

(2) For fixed $A$$A$AA$A$, in one of ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ or ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$, we have $H(A[i])=0$$H(A[i])=0$H(A[i])=0H(A[i])=0$H(A[i])=0$ if $i\ll 0$$i≪0$i≪0i \ll 0$i≪0$.

The finite functors are those for which we also have

(3) $H(A[i])=0$$H(A[i])=0$H(A[i])=0H(A[i])=0$H(A[i])=0$ for all $i\gg 0$$i≫0$i≫0i \gg 0$i≫0$.

Remark 6.8. The proof of part (1) of Theorem 6.6 may be found in [29], while the proof of part (2) of Theorem 6.6 occupies [28]. These are not easy theorems.

Let $\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$, with $X$$X$XX$X$ a scheme proper over a noetherian ring $R$$R$RR$R$. Then the hypotheses of Theorem 6.6(1) are satisfied. We learn (among other things) that the natural

1 What's important for the current survey is that, if $X$$X$XX$X$ is a noetherian, separated scheme, then $\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$ satisfies this hypothesis provided $X$$X$XX$X$ is finite-dimensional and quasiexcellent.
functor, taking an object $B\in {\mathbf{D}}_{\text{coh}}^b(X)$$B∈{\mathbf{D}}_{\text{coh }}^b(X)$B inD_("coh ")^(b)(X)B \in \mathbf{D}_{\text {coh }}^{b}(X)$B∈{\mathbf{D}}_{\text{coh }}^b(X)$ to the $R$$R$RR$R$-linear functor $\mathrm{Hom}(-,B):{\mathbf{D}}^{\text{perf}}(X{)}^{\mathrm{o}\mathrm{p}}\to$$\mathrm{Hom}⁡(−,B):{\mathbf{D}}^{\text{perf }}(X{)}^{\mathrm{o}\mathrm{p}}→$Hom(-,B):D^("perf ")(X)^(op)rarr\operatorname{Hom}(-, B): \mathbf{D}^{\text {perf }}(X)^{\mathrm{op}} \rightarrow$\mathrm{Hom}⁡(−,B):{\mathbf{D}}^{\text{perf }}(X{)}^{\mathrm{o}\mathrm{p}}→$ Mod- $R$$R$RR$R$, is a fully faithful embedding

whose essential image is precisely the finite homological functors.

If we further assume that the scheme $X$$X$XX$X$ is finite-dimensional and quasiexcellent then the hypotheses of Theorem 6.6(2) are also satisfied. We learn that the functor, taking an object $A\in {\mathbf{D}}^{\text{perf}}(X)$$A∈{\mathbf{D}}^{\text{perf }}(X)$A inD^("perf ")(X)A \in \mathbf{D}^{\text {perf }}(X)$A∈{\mathbf{D}}^{\text{perf }}(X)$ to the $R$$R$RR$R$-linear functor $\mathrm{Hom}(A,-)$$\mathrm{Hom}⁡(A,−)$Hom(A,-)\operatorname{Hom}(A,-)$\mathrm{Hom}⁡(A,−)$, is a fully faithful embedding

whose essential image is also the finite homological functors.

In [31, HISTORICAL SURVEY 8.2] the reader can find a discussion of the (algebrogeometric) precursors of Theorem 6.6. As for the applications, let us go through one of them.

Remark 6.9. Let $X$$X$XX$X$ be a scheme proper over the field $\mathbb{C}$$\mathbb{C}$C\mathbb{C}$\mathbb{C}$ of complex numbers, and let $X^{\text{an}}$$X^{\text{an }}$X^("an ")X^{\text {an }}$X^{\text{an }}$ be the underlying complex analytic space. The analytification induces a functor we will call $\mathcal{L}:{\mathbf{D}}_{\text{coh}}^b(X)\to {\mathbf{D}}_{\text{coh}}^b\left(X^{\text{an}}\right)$$\mathcal{L}:{\mathbf{D}}_{\text{coh }}^b(X)→{\mathbf{D}}_{\text{coh }}^b\left(X^{\text{an }}\right)$L:D_("coh ")^(b)(X)rarrD_("coh ")^(b)(X^("an "))\mathscr{L}: \mathbf{D}_{\text {coh }}^{b}(X) \rightarrow \mathbf{D}_{\text {coh }}^{b}\left(X^{\text {an }}\right)$\mathcal{L}:{\mathbf{D}}_{\text{coh }}^b(X)→{\mathbf{D}}_{\text{coh }}^b\left(X^{\text{an }}\right)$, it is the functor taking a bounded complex of coherent algebraic sheaves on $X$$X$XX$X$ to the analytification, which is a bounded complex of coherent analytic sheaves on $X^{\text{an}}$$X^{\text{an }}$X^("an ")X^{\text {an }}$X^{\text{an }}$. The pairing sending an object $A\in {\mathbf{D}}^{\text{perf}}(X)$$A∈{\mathbf{D}}^{\text{perf }}(X)$A inD^("perf ")(X)A \in \mathbf{D}^{\text {perf }}(X)$A∈{\mathbf{D}}^{\text{perf }}(X)$ and an object $B\in {\mathbf{D}}_{\text{coh}}^b\left(X^{\text{an}}\right)$$B∈{\mathbf{D}}_{\text{coh }}^b\left(X^{\text{an }}\right)$B inD_("coh ")^(b)(X^("an "))B \in \mathbf{D}_{\text {coh }}^{b}\left(X^{\text {an }}\right)$B∈{\mathbf{D}}_{\text{coh }}^b\left(X^{\text{an }}\right)$ to $\mathrm{Hom}(\mathcal{L}(A),B)$$\mathrm{Hom}⁡(\mathcal{L}(A),B)$Hom(L(A),B)\operatorname{Hom}(\mathscr{L}(A), B)$\mathrm{Hom}⁡(\mathcal{L}(A),B)$ delivers a map

Since the image lands in the finite homological functors, Theorem 6.6(1) allows us to factor this uniquely through the inclusion $\mathcal{Y}\circ i$$\mathcal{Y}∘i$Y@i\mathscr{Y} \circ i$\mathcal{Y}∘i$, that is, there exists (up to canonical natural isomorphism) a unique functor $\mathcal{R}$$\mathcal{R}$R\mathscr{R}$\mathcal{R}$ rendering commutative the triangle

And proving Serre's GAGA theorem reduces to the easy exercise of showing that $\mathcal{L}$$\mathcal{L}$L\mathscr{L}$\mathcal{L}$ and $\mathcal{R}$$\mathcal{R}$R\mathscr{R}$\mathcal{R}$ are inverse equivalences, the reader can find this in the (short) [29, SECTION 8 AND APPENDIX A].

The brilliant inspiration underpinning the approach is due to Jack Hall [12], he is the person who came up with the idea of using the pairing above, coupled with representability theorems, to prove GAGA. The representability theorems available to Jack Hall at the time were not powerful enough, and Theorem 6.6 was motivated by trying to find a direct path from the ingenious, simple idea to a fullblown proof.

Discussion 6.10. In preparation for the next theorem, we give a very brief review of metrics in triangulated categories. The reader is referred to the survey article [30] for a much fuller and more thorough account.

Given a triangulated category $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$, a metric on $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ assigns a length to every morphism. In this article the only metrics we consider are the ones arising from t-structures. If $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is an approximable triangulated category we choose a t-structure $(\mathcal{T}\le 0,{\mathcal{T}}^0)$$\left(\mathcal{T}≤0,{\mathcal{T}}^0\right)$(T <= 0,T^(0))\left(\mathscr{T} \leq 0, \mathscr{T}^{0}\right)$(\mathcal{T}≤0,{\mathcal{T}}^0)$ in the preferred equivalence class, and this induces a metric as follows. Given a morphism $f:X\to Y$$f:X→Y$f:X rarr Yf: X \rightarrow Y$f:X→Y$, we may complete to an exact triangle $X\stackrel{f}{\to }Y\to D$$X\stackrel{f}{→}Y→D$Xrarr"f"Y rarr DX \xrightarrow{f} Y \rightarrow D$X\stackrel{f}{→}Y→D$, and the length of $f$$f$ff$f$ is given by the formula

In this survey we allow the length of a morphism to be infinite; if the set on the right is empty then we declare Length $(f)=\mathrm{\infty }$$(f)=\mathrm{∞}$(f)=oo(f)=\infty$(f)=\mathrm{∞}$.

This metric depends on the choice of a $t$$t$tt$t$-structure, but not a lot. As all $t$$t$tt$t$-structures in the preferred equivalence class are equivalent, any two preferred t-structures will give rise to equivalent metrics (with an obvious definition of equivalence of metrics).

Note that if $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is a triangulated category and $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ is a triangulated subcategory, then a metric on $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ restricts to a metric on $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$. In particular, if $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is approximable, the metric on $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ of the previous paragraph restricts to give metrics on the full subcategories ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ and ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$. Once again these metrics are only defined up to equivalence. And, of course, a metric on $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ is also a metric on ${\mathcal{S}}^{\text{op}}$${\mathcal{S}}^{\text{op }}$S^("op ")\mathscr{S}^{\text {op }}${\mathcal{S}}^{\text{op }}$, thus we have specified (up to equivalence) canonical metrics on ${\mathcal{T}}^c,{\mathcal{T}}_c^b,{\left[{\mathcal{T}}^c\right]}^{\text{op}}$${\mathcal{T}}^c,{\mathcal{T}}_c^b,{\left[{\mathcal{T}}^c\right]}^{\text{op }}$T^(c),T_(c)^(b),[T^(c)]^("op ")\mathscr{T}^{c}, \mathscr{T}_{c}^{b},\left[\mathscr{T}^{c}\right]^{\text {op }}${\mathcal{T}}^c,{\mathcal{T}}_c^b,{\left[{\mathcal{T}}^c\right]}^{\text{op }}$, and ${\left[{\mathcal{T}}_c^b\right]}^{\text{op}}$${\left[{\mathcal{T}}_c^b\right]}^{\text{op }}$[T_(c)^(b)]^("op ")\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}${\left[{\mathcal{T}}_c^b\right]}^{\text{op }}$.

Suppose $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ is a triangulated category with a metric. A Cauchy sequence in $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ is a sequence of morphisms $E_1\to E_2\to E_3\to \cdots$$E_1→E_2→E_3→⋯$E_(1)rarrE_(2)rarrE_(3)rarr cdotsE_{1} \rightarrow E_{2} \rightarrow E_{3} \rightarrow \cdots$E_1→E_2→E_3→⋯$ which eventually become arbitrarily short.

embeds $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ into the category Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ of additive functors ${\mathcal{S}}^{\text{op}}\to \mathcal{A}\ell$${\mathcal{S}}^{\text{op }}→\mathcal{A}\mathrm{ℓ}$S^("op ")rarrAℓ\mathscr{S}^{\text {op }} \rightarrow \mathscr{A} \mathscr{\ell}${\mathcal{S}}^{\text{op }}→\mathcal{A}\mathrm{ℓ}$. In the category Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ colimits exist, allowing us to define

(1) The category $\mathfrak{L}(\mathcal{S})$$\mathfrak{L}(\mathcal{S})$L(S)\mathfrak{L}(\mathscr{S})$\mathfrak{L}(\mathcal{S})$ is the full subcategory of Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$, whose objects are the colimits of Yoneda images of Cauchy sequences in $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$;

(2) The full subcategory $\mathcal{S}(\mathcal{S})\subset \mathbb{R}(\mathcal{S})$$\mathcal{S}(\mathcal{S})⊂\mathbb{R}(\mathcal{S})$S(S)subR(S)\mathscr{S}(\mathscr{S}) \subset \mathbb{R}(\mathscr{S})$\mathcal{S}(\mathcal{S})⊂\mathbb{R}(\mathcal{S})$ has for objects those functors $F\in \mathbb{R}(\mathcal{S})\subset$$F∈\mathbb{R}(\mathcal{S})⊂$F inR(S)subF \in \mathbb{R}(\mathscr{S}) \subset$F∈\mathbb{R}(\mathcal{S})⊂$ Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ which take sufficiently short morphisms to isomorphisms. In symbols, $F\in \mathfrak{L}(\mathcal{S})$$F∈\mathfrak{L}(\mathcal{S})$F inL(S)F \in \mathfrak{L}(\mathscr{S})$F∈\mathfrak{L}(\mathcal{S})$ belongs to $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ if there exists an $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$ such that

(3) The exact triangles in $\mathbb{S}(\mathcal{S})$$\mathbb{S}(\mathcal{S})$S(S)\mathbb{S}(\mathscr{S})$\mathbb{S}(\mathcal{S})$ are the colimits in Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ of Yoneda images of Cauchy sequences of exact triangles in $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$, where the colimits happen to lie in $\subseteq (\mathcal{S})$$⊆(\mathcal{S})$sube(S)\subseteq(\mathscr{S})$⊆(\mathcal{S})$.

A word of caution about (3): if we are given in $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ a Cauchy sequence of exact triangles, we can form the colimit in Mod- $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ of its Yoneda image. This colimit is guaranteed to lie in $\mathfrak{R}(\mathcal{S})$$\mathfrak{R}(\mathcal{S})$R(S)\mathfrak{R}(\mathscr{S})$\mathfrak{R}(\mathcal{S})$, but will not usually lie in the smaller $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$. If it happens to lie in $\mathbb{S}(\mathcal{S})$$\mathbb{S}(\mathcal{S})$S(S)\mathbb{S}(\mathscr{S})$\mathbb{S}(\mathcal{S})$ then (3) declares it to be an exact triangle in $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$.

And now we are ready for the theorem.

Theorem 6.11. Let $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ be a triangulated category with a metric. Assume the metric is good; this is a technical term, see [30, DEFInITION 10] for the precise formulation. Then

(1) The category $\mathrm{\Im }(\mathcal{S})$$\mathrm{â„‘}(\mathcal{S})$â„‘(S)\mathbb{\Im}(\mathscr{S})$\mathrm{â„‘}(\mathcal{S})$ of Discussion 6.10(2), with the exact triangles as defined in Discussion 6.10(3), is a triangulated category.

Now let $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ be an approximable triangulated category. In Discussion 6.10 we constructed (up to equivalence) a metric on $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$, and hence on its subcategories ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ and ${\left[{\mathcal{T}}_c^b\right]}^{\mathrm{o}\mathrm{p}}$${\left[{\mathcal{T}}_c^b\right]}^{\mathrm{o}\mathrm{p}}$[T_(c)^(b)]^(op)\left[\mathscr{T}_{c}^{b}\right]^{\mathrm{op}}${\left[{\mathcal{T}}_c^b\right]}^{\mathrm{o}\mathrm{p}}$. Those metrics are all good, and the theorem goes on to give natural, exact equivalences of triangulated categories

(2) $\mathbb{S}\left({\mathcal{T}}^c\right)\cong {\mathcal{T}}_c^b$$\mathbb{S}\left({\mathcal{T}}^c\right)â‰\dots {\mathcal{T}}_c^b$S(T^(c))~=T_(c)^(b)\mathbb{S}\left(\mathscr{T}^{c}\right) \cong \mathscr{T}_{c}^{b}$\mathbb{S}\left({\mathcal{T}}^c\right)â‰\dots {\mathcal{T}}_c^b$. This equivalence is unconditional.

(3) If the approximable triangulated category $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ happens to be noetherian as in [27, DEFINITION 5.1], then $\mathrm{\Im }\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op}}\right)\cong {\left[{\mathcal{T}}^c\right]}^{\text{op}}$$\mathrm{â„‘}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)â‰\dots {\left[{\mathcal{T}}^c\right]}^{\text{op }}$â„‘([T_(c)^(b)]^("op "))~=[T^(c)]^("op ")\Im\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right) \cong\left[\mathscr{T}^{c}\right]^{\text {op }}$\mathrm{â„‘}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)â‰\dots {\left[{\mathcal{T}}^c\right]}^{\text{op }}$.

Remark 6.12. First of all, in Theorem 6.11(3) we assumed that the approximable triangulated $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is noetherian as in [27, DEFInITIon 5.1]. The only observation we want to make here is that if $X$$X$XX$X$ is a noetherian, separated scheme then the approximable triangulated category $\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$ is noetherian. Thus, for noetherian, separated schemes $X$$X$XX$X$, Theorem 6.11 gives exact equivalences of triangulated categories

The research paper [27] contains the proofs of the assertions in Theorem 6.11. The reader can find a skimpy survey in [31, SECTION 9] and a more extensive one in [30]. In [31, HISTORICAL SURVEY 9.1] there is a discussion of precursors of the results.

New scientific developments are tentative and unpolished; only with the passage of time do they acquire the gloss and elegance of a refined, varnished theory. And there is nothing more difficult to predict than the future. My colleague Neil Trudinger used to joke that my beard makes me look like a biblical prophet-the reader should not be deceived, appearances are notoriously misleading, the abundance of facial hair is not a reliable yardstick for measuring the gift of foresight that marks out a visionary, and I am certifiably not a clairvoyant. All I do in this section is offer a handful of obvious questions that spring to mind. The list is not meant to be exhaustive, and might well be missing major tableaux of the overall picture. It is entirely possible that the future will see this theory flourish in directions orthogonal to those sketched here.

Let us begin with what is freshest in our minds: we have just seen Theorem 6.11, part (1) of which tells us that, given a triangulated category $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ with a good metric, there is a recipe producing another triangulated category $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$, which, as it happens, comes with an induced good metric. We can ask:

Problem 7.1. Can one formulate reasonable sufficient conditions, on the triangulated category $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ and on its good metric, to guarantee that $\mathrm{\Im }(S(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}})={\mathcal{S}}^{\mathrm{o}\mathrm{p}}$$\mathrm{â„‘}\left(S(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}}\right)={\mathcal{S}}^{\mathrm{o}\mathrm{p}}$â„‘(S(S)^(op))=S^(op)\Im\left(S(\mathscr{S})^{\mathrm{op}}\right)=\mathscr{S}^{\mathrm{op}}$\mathrm{â„‘}(S(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}})={\mathcal{S}}^{\mathrm{o}\mathrm{p}}$ ? Who knows, maybe even necessary and sufficient conditions?

Motivating Example 7.2. Let $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ be an approximable triangulated category and put $\mathcal{S}={\mathcal{T}}^c$$\mathcal{S}={\mathcal{T}}^c$S=T^(c)\mathscr{S}=\mathscr{T}^{c}$\mathcal{S}={\mathcal{T}}^c$, with the metric of Discussion 6.10. Theorem 6.11(2) computes for us that $\mathbb{S}\left({\mathcal{T}}^c\right)\cong {\mathcal{T}}_c^b$$\mathbb{S}\left({\mathcal{T}}^c\right)â‰\dots {\mathcal{T}}_c^b$S(T^(c))~=T_(c)^(b)\mathbb{S}\left(\mathscr{T}^{c}\right) \cong \mathscr{T}_{c}^{b}$\mathbb{S}\left({\mathcal{T}}^c\right)â‰\dots {\mathcal{T}}_c^b$. I ask the reader to believe that the natural, induced metric on $₫\left({\mathcal{T}}^c\right)$$â‚«\left({\mathcal{T}}^c\right)$â‚«(T^(c))â‚«\left(\mathscr{T}^{c}\right)$â‚«\left({\mathcal{T}}^c\right)$ agrees with the metric on ${\mathcal{T}}_c^b\subset \mathcal{T}$${\mathcal{T}}_c^b⊂\mathcal{T}$T_(c)^(b)subT\mathscr{T}_{c}^{b} \subset \mathscr{T}${\mathcal{T}}_c^b⊂\mathcal{T}$ given in Discussion 6.10. Now Theorem 6.11(3) goes on to tell us that, as long as the approximable triangulated category $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is noetherian, we also have that $\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op}}\right)\cong {\left[{\mathcal{T}}^c\right]}^{\text{op}}$$\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)â‰\dots {\left[{\mathcal{T}}^c\right]}^{\text{op }}$S([T_(c)^(b)]^("op "))~=[T^(c)]^("op ")\mathbb{S}\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right) \cong\left[\mathscr{T}^{c}\right]^{\text {op }}$\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)â‰\dots {\left[{\mathcal{T}}^c\right]}^{\text{op }}$; as it happens, the induced good metric on $\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op}}\right)$$\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)$S([T_(c)^(b)]^("op "))\mathbb{S}\left(\left[\mathscr{T}_{c}^{b}\right]^{\text {op }}\right)$\mathbb{S}\left({\left[{\mathcal{T}}_c^b\right]}^{\text{op }}\right)$ also agrees, up to equivalence, with the metric that Discussion 6.10 created on ${\left[{\mathcal{T}}^c\right]}^{\text{op}}$${\left[{\mathcal{T}}^c\right]}^{\text{op }}$[T^(c)]^("op ")\left[\mathscr{T}^{c}\right]^{\text {op }}${\left[{\mathcal{T}}^c\right]}^{\text{op }}$. Combining these we have many examples of exact equivalences of triangulated categories $\mathbb{S}(\mathbb{S}(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}})\cong {\mathcal{S}}^{\mathrm{o}\mathrm{p}}$$\mathbb{S}\left(\mathbb{S}(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}}\right)â‰\dots {\mathcal{S}}^{\mathrm{o}\mathrm{p}}$S(S(S)^(op))~=S^(op)\mathbb{S}\left(\mathbb{S}(\mathscr{S})^{\mathrm{op}}\right) \cong \mathscr{S}^{\mathrm{op}}$\mathbb{S}(\mathbb{S}(\mathcal{S}{)}^{\mathrm{o}\mathrm{p}})â‰\dots {\mathcal{S}}^{\mathrm{o}\mathrm{p}}$, which are homeomorphisms with respect to the metrics. Thus Problem 7.1 asks the reader to find the right generalization.

Next one can wonder about the functoriality of the construction. Suppose $F:\mathcal{S}\to \mathcal{T}$$F:\mathcal{S}→\mathcal{T}$F:SrarrTF: \mathscr{S} \rightarrow \mathscr{T}$F:\mathcal{S}→\mathcal{T}$ is a triangulated functor, and that both $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ and $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ have good metrics. What are reasonable sufficient conditions which guarantee the existence of an induced functor $\mathrm{\Xi }(F)$$\mathrm{Ξ}(F)$Xi(F)\Xi(F)$\mathrm{Ξ}(F)$, either from $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ to $\mathcal{S}(\mathcal{T})$$\mathcal{S}(\mathcal{T})$S(T)\mathscr{S}(\mathscr{T})$\mathcal{S}(\mathcal{T})$ or in the other direction? So far there is one known result of this genre, the reader can find the statement below in Sun and Zhang [37, THEOREM 1.1(3)].

Theorem 7.3. Suppose we are given two triangulated categories $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ and $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$, both with good metrics. Suppose we are also given a pair of functors $F:\mathcal{S}\rightleftarrows \mathcal{T}:G$$F:\mathcal{S}⇄\mathcal{T}:G$F:S⇄T:GF: \mathscr{S} \rightleftarrows \mathscr{T}: G$F:\mathcal{S}⇄\mathcal{T}:G$ with $F⊣G$$FâŠ\pounds G$F⊣GF \dashv G$FâŠ\pounds G$, meaning that $F$$F$FF$F$ is left adjoint to $G$$G$GG$G$. Assume further that both $F$$F$FF$F$ and $G$$G$GG$G$ are continuous with respect to the metrics, in the obvious sense.

Then the functor $\hat{F}:\mathrm{Mod}-\mathcal{T}\to \mathrm{Mod}-\mathcal{S}$$\widehat{F}:\mathrm{Mod}−\mathcal{T}→\mathrm{Mod}−\mathcal{S}$hat(F):Mod-Trarr Mod-S\hat{F}: \operatorname{Mod}-\mathscr{T} \rightarrow \operatorname{Mod}-\mathscr{S}$\hat{F}:\mathrm{Mod}−\mathcal{T}→\mathrm{Mod}−\mathcal{S}$ induced by composition with $F$$F$FF$F$, that is,

the functor taking the $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$-module $H:{\mathcal{T}}^{\mathrm{o}\mathrm{p}}\to \mathcal{A}b$$H:{\mathcal{T}}^{\mathrm{o}\mathrm{p}}→\mathcal{A}b$H:T^(op)rarrAbH: \mathscr{T}^{\mathrm{op}} \rightarrow \mathscr{A} b$H:{\mathcal{T}}^{\mathrm{o}\mathrm{p}}→\mathcal{A}b$ to the $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$-module $(H\circ F):{\mathcal{S}}^{\mathrm{o}\mathrm{p}}\to \mathcal{A}b$$(H∘F):{\mathcal{S}}^{\mathrm{o}\mathrm{p}}→\mathcal{A}b$(H@F):S^(op)rarrAb(H \circ F): \mathscr{S}^{\mathrm{op}} \rightarrow \mathscr{A} b$(H∘F):{\mathcal{S}}^{\mathrm{o}\mathrm{p}}→\mathcal{A}b$, restricts to a functor which we will denote $\mathrm{\Xi }(F):\subseteq (\mathcal{T})\to \mathrm{\Im }(\mathcal{S})$$\mathrm{Ξ}(F):⊆(\mathcal{T})→\mathrm{ℑ}(\mathcal{S})$Xi(F):⊆(T)rarrℑ(S)\Xi(F): \subseteq(\mathscr{T}) \rightarrow \mathbb{\Im}(\mathscr{S})$\mathrm{Ξ}(F):⊆(\mathcal{T})→\mathrm{ℑ}(\mathcal{S})$. That is, the functor $\mathrm{\Xi }(F)$$\mathrm{Ξ}(F)$Xi(F)\Xi(F)$\mathrm{Ξ}(F)$ is defined to be the unique map making the square below commute

where the vertical inclusions are given by the definition of $\mathbb{S}(?)\subset \mathbb{R}(?)\subset$$\mathbb{S}(?)⊂\mathbb{R}(?)⊂$S(?)subR(?)sub\mathbb{S}(?) \subset \mathbb{R}(?) \subset$\mathbb{S}(?)⊂\mathbb{R}(?)⊂$ Mod-? of Discus sion 6.10 (1) and (2).

Furthermore, the functor $\subseteq (F)$$⊆(F)$sube(F)\subseteq(F)$⊆(F)$ respects the exact triangles as defined in Discussion $6.10(3)$$6.10(3)$6.10(3)6.10(3)$6.10(3)$.

Sun and Zhang go on to study recollements. Suppose we are given a recollement of triangulated categories

If all three triangulated categories come with good metrics, and if all six functors are continuous, then the following may be found in [37, THEOREM 1.2].

Theorem 7.4. Under the hypotheses above, applying $\mathbb{S}$$\mathbb{S}$S\mathbb{S}$\mathbb{S}$ yields a right recollement

In the presence of enough continuous adjoints, we deduce that a semiorthogonal decomposition of $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ gives rise to a semiorthogonal decomposition of $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$. In view of the fact that there are metrics on ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ and ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$ such that

it is natural to wonder how the recent theorem of Sun and Zhang [37, THEOREM 1.2] compares with the older work of Kuznetsov [19, SECTION 2.5] and [20, SECTION 4].

The above shows that, subject to suitable hypotheses, the construction taking $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ to $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ can preserve (some of) the internal structure on the category $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$-for example, semiorthogonal decompositions. This leads naturally to

Problem 7.5. What other pieces of the internal structure of $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ are respected by the construction that passes to $\subseteq (\mathcal{S})$$⊆(\mathcal{S})$sube(S)\subseteq(\mathscr{S})$⊆(\mathcal{S})$ ? Under what conditions are these preserved?

Problem 7.5 may sound vague, but it can be made precise enough. For example, there is a huge literature dealing with the group of autoequivalences of the derived categories ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$. Now, as it happens, the metrics for which Remark 6.12 gives the equivalences

can be given (up to equivalence) intrinsic descriptions. Note that the way we introduced these metrics, in Discussion 6.10, was to use a preferred t-structure on $\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)$\mathcal{T}={\mathbf{D}}_{\mathbf{q}\mathbf{c}}(X)$ to give on $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ a metric, unique up to equivalence, and hence induced metrics on ${\mathcal{T}}^c={\mathbf{D}}^{\text{perf}}(X)$${\mathcal{T}}^c={\mathbf{D}}^{\text{perf }}(X)$T^(c)=D^("perf ")(X)\mathscr{T}^{c}=\mathbf{D}^{\text {perf }}(X)${\mathcal{T}}^c={\mathbf{D}}^{\text{perf }}(X)$ and on ${\mathcal{T}}_c^b={\mathbf{D}}_{\text{coh}}^b(X)$${\mathcal{T}}_c^b={\mathbf{D}}_{\text{coh }}^b(X)$T_(c)^(b)=D_("coh ")^(b)(X)\mathscr{T}_{c}^{b}=\mathbf{D}_{\text {coh }}^{b}(X)${\mathcal{T}}_c^b={\mathbf{D}}_{\text{coh }}^b(X)$ which are also unique up to equivalence. But this description seems to depend on an embedding into the large category $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$. What I am asserting now is that there are alternative descriptions of the same equivalence classes of metrics on ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ and on ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$, which do not use the embedding into $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$. The interested reader can find this in the later sections of [27]. Anyway, a consequence is that any autoequivalence, of either ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ or of ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$, must be continuous with a continuous inverse. Hence the group of autoequivalences of ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$ must be isomorphic to the group of autoequivalences of ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$. Or more generally, assume $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is a noetherian, approximable triangulated category, where noetherian has the meaning of [27, DEFINITION 5.1]. Then the group of exact autoequivalences of ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ is canonically isomorphic to the group of exact autoequivalences of ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$.

Are there similar theorems about t-structures in $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ going to t-structures in $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ ? Or about stability conditions on $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ mapping to stability conditions on $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ ?

We should note that any such theorem will have to come with conditions. After all, the category ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$ always has a bounded t-structure, while Antieau, Gepner, and Heller [3,

THEOREM 1.1] show that ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ does not in general. Thus it is possible for $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ to have a bounded t-structure but for $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$ not to. And in this particular example, the equivalence class of the metric has an intrinsic description, in the sense mentioned above.

Perhaps we should remind the reader that the article [3], by Antieau, Gepner, and Heller, finds a $K$$K$KK$K$-theoretic obstruction to the existence of bounded t-structures, more precisely if an appropriate category $\mathcal{E}$$\mathcal{E}$E\mathscr{E}$\mathcal{E}$ has a bounded t-structure then $K_{-1}(\mathcal{E})=0$$K_{−1}(\mathcal{E})=0$K_(-1)(E)=0K_{-1}(\mathscr{E})=0$K_{−1}(\mathcal{E})=0$. Hence the reference to [3] immediately raises the question of how the construction passing from $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ to $\mathbb{S}(\mathcal{S})$$\mathbb{S}(\mathcal{S})$S(S)\mathbb{S}(\mathscr{S})$\mathbb{S}(\mathcal{S})$ might relate to $K$$K$KK$K$-theory, especially to negative $K$$K$KK$K$-theory. Of course, one has to be a little circumspect here. While there is a $K$$K$KK$K$-theory for triangulated categories (see [25] for a survey), this $K$$K$KK$K$-theory has only been proved to behave well for "nice" triangulated categories, for example, for triangulated categories with bounded t-structures. Invariants like negative $K$$K$KK$K$-theory have never been defined for triangulated categories, and might well give nonsense. In what follows we will assume that all the $K$$K$KK$K$-theoretic statements are for triangulated categories with chosen enhancements, and that $K$$K$KK$K$-theory means the Waldhausen $K$$K$KK$K$-theory of the enhancement. We recall in passing that the enhancements are unique for many interesting classes of triangulated categories, see Lunts and Orlov [21], Canonaco and Stellari [9], Antieau [2] and Canonaco, Neeman, and Stellari [8].

With the disclaimers out of the way, what do the results surveyed in this article have to do with negative $K$$K$KK$K$-theory?

Let us begin with Schlichting's conjecture [36, CONJEcTURE 1 OF SECTION 10]; this conjecture, now known to be false [32], predicted that the negative $K$$K$KK$K$-theory of any abelian category should vanish. But Schlichting also proved that (1) $K_{-1}(\mathcal{A})=0$$K_{−1}(\mathcal{A})=0$K_(-1)(A)=0K_{-1}(\mathscr{A})=0$K_{−1}(\mathcal{A})=0$ for any abelian category $\mathcal{A}$$\mathcal{A}$A\mathscr{A}$\mathcal{A}$, and (2) $K_{-n}(\mathcal{A})=0$$K_{−n}(\mathcal{A})=0$K_(-n)(A)=0K_{-n}(\mathscr{A})=0$K_{−n}(\mathcal{A})=0$ whenever $\mathcal{A}$$\mathcal{A}$A\mathscr{A}$\mathcal{A}$ is a noetherian abelian category and $n>0$$n>0$n > 0n>0$n>0$. Now note that the $K(\mathcal{A})=K\left({\mathcal{A}}^{\mathrm{o}\mathrm{p}}\right)$$K(\mathcal{A})=K\left({\mathcal{A}}^{\mathrm{o}\mathrm{p}}\right)$K(A)=K(A^(op))K(\mathscr{A})=K\left(\mathscr{A}^{\mathrm{op}}\right)$K(\mathcal{A})=K\left({\mathcal{A}}^{\mathrm{o}\mathrm{p}}\right)$, hence the negative $K$$K$KK$K$-theory of any artinian abelian category must also vanish. And playing with extensions of abelian categories, we easily deduce the vanishing of the negative $K$$K$KK$K$-theory of a sizeable class of abelian categories. So while Schlichting's conjecture is false in the generality in which it was stated, there is some large class of abelian categories for which it is true. The challenge is to understand this class.

It becomes interesting to see what relation, if any, the results surveyed here have with this question.

Let us begin with Theorems 4.4 and 4.2. Theorem 4.4 tells us that, when $X$$X$XX$X$ is a

quasiexcellent, finite-dimensional, separated noetherian scheme, the category ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$ is strongly generated. This category has a unique enhancement whose $K$$K$KK$K$-theory agrees with the $K$$K$KK$K$-theory of the noetherian abelian category $\mathrm{Coh}(X)$$\mathrm{Coh}⁡(X)$Coh(X)\operatorname{Coh}(X)$\mathrm{Coh}⁡(X)$, hence the negative $K$$K$KK$K$-theory vanishes. Theorem 4.2 and Remark 4.3 tell us that the category ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ has a strong generator if and only if $X$$X$XX$X$ is regular and finite-dimensional-in which case it is equivalent to ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$ and its unique enhancement has vanishing negative $K$$K$KK$K$-theory. This raises the question:

Problem 7.6. If $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is a triangulated category with a strong generator, does it follow that any enhancement of $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ has vanishing negative $K$$K$KK$K$-theory?

Let us refine this question a little. In Definition 4.1 we learned that a strong generator, for a triangulated category $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$, is an object $G\in \mathcal{T}$$G∈\mathcal{T}$G inTG \in \mathscr{T}$G∈\mathcal{T}$ such that there exists an integer $\ell >0$$\mathrm{ℓ}>0$ℓ > 0\ell>0$\mathrm{ℓ}>0$
with $\mathcal{T}=⟨G{⟩}_{\ell }$$\mathcal{T}=⟨G{âŸ\copyright }_{\mathrm{â„“}}$T=(:G:)_(â„“)\mathscr{T}=\langle G\rangle_{\ell}$\mathcal{T}=⟨G{âŸ\copyright }_{\mathrm{â„“}}$. Following Rouquier, we can ask for estimates on the integer $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$. This leads us to:

Definition 7.7. Let $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ be a triangulated category. The Rouquier dimension of $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ is the smallest integer $\ell \ge 0$$\mathrm{â„“}â‰\yen 0$â„“ >= 0\ell \geq 0$\mathrm{â„“}â‰\yen 0$ (we allow the possibility $\ell =\mathrm{\infty }$$\mathrm{â„“}=\mathrm{∞}$â„“=oo\ell=\infty$\mathrm{â„“}=\mathrm{∞}$ ), for which there exists an object $G$$G$GG$G$ with $\mathcal{T}=⟨G{⟩}_{\ell +1}$$\mathcal{T}=⟨G{âŸ\copyright }_{\mathrm{â„“}+1}$T=(:G:)_(â„“+1)\mathscr{T}=\langle G\rangle_{\ell+1}$\mathcal{T}=⟨G{âŸ\copyright }_{\mathrm{â„“}+1}$. See Rouquier [35] for much more about this fascinating invariant.

There is a rich and beautiful literature estimating this invariant and its various cousins-see Rouquier [35] for the origins of the theory, and a host of other places for subsequent developments. For this survey we note only that, for ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$, the Rouquier dimension is conjectured to be equal to the Krull dimension of $X$$X$XX$X$. But by a conjecture of Weibel [39], now a theorem of Kerz, Strunk, and Tamme [18], the Krull dimension of $X$$X$XX$X$ also has a $K$$K$KK$K$ theoretic description: the groups $K_n$$K_n$K_(n)K_{n}$K_n$ of the unique enhancement of ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ vanish for all $n<-\dim (X)$$n<−\dim ⁡(X)$n < -dim(X)n<-\operatorname{dim}(X)$n<−\dim ⁡(X)$. Recalling that $\mathcal{S}={\mathbf{D}}_{\text{coh}}^b(X)$$\mathcal{S}={\mathbf{D}}_{\text{coh }}^b(X)$S=D_("coh ")^(b)(X)\mathscr{S}=\mathbf{D}_{\text {coh }}^{b}(X)$\mathcal{S}={\mathbf{D}}_{\text{coh }}^b(X)$ is related to ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ by the fact that the construction $\mathbb{S}$$\mathbb{S}$S\mathbb{S}$\mathbb{S}$ interchanges them (up to passing to opposite categories, which has no effect on $K$$K$KK$K$-theory), this leads us to ask:

Problem 7.8. Let $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ be a regular (= strongly generated) triangulated category as in Definition 4.1, and let $N<\mathrm{\infty }$$N<\mathrm{∞}$N < ooN<\infty$N<\mathrm{∞}$ be its Rouquier dimension. Is it true that $K_n$$K_n$K_(n)K_{n}$K_n$ vanishes on any enhancement of $⋐(\mathcal{S})$$⋐(\mathcal{S})$⋐(S)\Subset(\mathscr{S})$⋐(\mathcal{S})$, for any metric on $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ and whenever $n<-N$$n<−N$n < -Nn<-N$n<−N$ ?

In an entirely different direction, we know that the construction $\mathbb{S}$$\mathbb{S}$S\mathbb{S}$\mathbb{S}$ interchanges ${\mathbf{D}}^{\text{perf}}(X)$${\mathbf{D}}^{\text{perf }}(X)$D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)${\mathbf{D}}^{\text{perf }}(X)$ and ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$, and that these categories coincide if and only if $X$$X$XX$X$ is regular. This leads us to ask:

Problem 7.9. Is there a way to measure the "distance" between $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ and $\mathcal{S}(\mathcal{S})$$\mathcal{S}(\mathcal{S})$S(S)\mathscr{S}(\mathscr{S})$\mathcal{S}(\mathcal{S})$, in such a way that resolution of singularities can be viewed as a process reducing this distance? Who knows, maybe there is even a good metric on $\mathcal{S}={\mathbf{D}}^{\text{perf}}(X)$$\mathcal{S}={\mathbf{D}}^{\text{perf }}(X)$S=D^("perf ")(X)\mathscr{S}=\mathbf{D}^{\text {perf }}(X)$\mathcal{S}={\mathbf{D}}^{\text{perf }}(X)$ and/or on ${\mathcal{S}}^{\mathrm{\prime }}={\mathbf{D}}_{\text{coh}}^b(X)$${\mathcal{S}}^{\mathrm{′}}={\mathbf{D}}_{\text{coh }}^b(X)$S^(')=D_("coh ")^(b)(X)\mathscr{S}^{\prime}=\mathbf{D}_{\text {coh }}^{b}(X)${\mathcal{S}}^{\mathrm{′}}={\mathbf{D}}_{\text{coh }}^b(X)$ such that the construction $₫$$â‚«$â‚«â‚«$â‚«$ takes either $\mathcal{S}$$\mathcal{S}$S\mathscr{S}$\mathcal{S}$ or ${\mathcal{S}}^{\mathrm{\prime }}$${\mathcal{S}}^{\mathrm{′}}$S^(')\mathscr{S}^{\prime}${\mathcal{S}}^{\mathrm{′}}$ to an $\subseteq (\mathcal{S})$$⊆(\mathcal{S})$sube(S)\subseteq(\mathscr{S})$⊆(\mathcal{S})$ or $₫\left({\mathcal{S}}^{\mathrm{\prime }}\right)$$â‚«\left({\mathcal{S}}^{\mathrm{′}}\right)$â‚«(S^('))â‚«\left(\mathscr{S}^{\prime}\right)$â‚«\left({\mathcal{S}}^{\mathrm{′}}\right)$ which is ${\mathbf{D}}^{\text{perf}}(Y)={\mathbf{D}}_{\text{coh}}^b(Y)$${\mathbf{D}}^{\text{perf }}(Y)={\mathbf{D}}_{\text{coh }}^b(Y)$D^("perf ")(Y)=D_("coh ")^(b)(Y)\mathbf{D}^{\text {perf }}(Y)=\mathbf{D}_{\text {coh }}^{b}(Y)${\mathbf{D}}^{\text{perf }}(Y)={\mathbf{D}}_{\text{coh }}^b(Y)$ for some resolution of singularities $Y$$Y$YY$Y$ of $X$$X$XX$X$.

While on the subject of regularity ( $=$$=$==$=$ strong generation):

Problem 7.10. Is there some way to understand which are the approximable triangulated categories $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ for which ${\mathcal{T}}^c$${\mathcal{T}}^c$T^(c)\mathscr{T}^{c}${\mathcal{T}}^c$ and/or ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$ are regular?

Theorems 4.2 and 4.4 deal with the case $\mathcal{T}={\mathbf{D}}_{\text{qc}}(X)$$\mathcal{T}={\mathbf{D}}_{\text{qc }}(X)$T=D_("qc ")(X)\mathscr{T}=\mathbf{D}_{\text {qc }}(X)$\mathcal{T}={\mathbf{D}}_{\text{qc }}(X)$. Approximability is used in the proofs given in [33] and [4], but only to ultimately reduce to the case of ${\mathcal{T}}^c={\mathbf{D}}^{\text{perf}}(X)$${\mathcal{T}}^c={\mathbf{D}}^{\text{perf }}(X)$T^(c)=D^("perf ")(X)\mathscr{T}^{c}=\mathbf{D}^{\text {perf }}(X)${\mathcal{T}}^c={\mathbf{D}}^{\text{perf }}(X)$ with $X$$X$XX$X$ an affine scheme-this case turns out to be classical, it was settled already in Kelly's 1965 article [17]. And the diverse precursors of Theorems 4.2 and 4.4, which we have hardly mentioned in the current survey, are also relatively narrow in scope. But presumably there are other proofs out there, yet to be discovered. And new approaches might well lead to generalizations that hold for triangulated categories having nothing to do with algebraic geometry.

Next let us revisit Theorem 6.6, the theorem identifying each of ${\left[{\mathcal{T}}^c\right]}^{\text{op}}$${\left[{\mathcal{T}}^c\right]}^{\text{op }}$[T^(c)]^("op ")\left[\mathscr{T}^{c}\right]^{\text {op }}${\left[{\mathcal{T}}^c\right]}^{\text{op }}$ (respectively ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$ ) as the finite homological functors on the other. In view of the motivating application, discussed in Remark 6.9, and of the generality of Theorem 6.6, it is natural to wonder:

Problem 7.11. Do GAGA-type theorems have interesting generalizations to other approximable triangulated categories? The reader is invited to check [29, SECTION 8 AND APPENDIX A]: except for the couple of paragraphs in [29, EXAMPLE A.2] everything is formulated in gorgeous generality and might be applicable in other contexts.

In the context of ${\mathbf{D}}_{\text{coh}}^b(X)$${\mathbf{D}}_{\text{coh }}^b(X)$D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)${\mathbf{D}}_{\text{coh }}^b(X)$, where $X$$X$XX$X$ is a scheme proper over a noetherian ring $R$$R$RR$R$, there was a wealth of different-looking GAGA-statements before Jack Hall's lovely paper [12] unified them into one. In other words, the category ${\mathbf{D}}_{\text{coh}}^b(X)={\mathcal{T}}_c^b$${\mathbf{D}}_{\text{coh }}^b(X)={\mathcal{T}}_c^b$D_("coh ")^(b)(X)=T_(c)^(b)\mathbf{D}_{\text {coh }}^{b}(X)=\mathscr{T}_{c}^{b}${\mathbf{D}}_{\text{coh }}^b(X)={\mathcal{T}}_c^b$ had many differentlooking incarnations, and it was not until Hall's paper that it was understood that there was one underlying reason why they all coincided.

Hence Problem 7.11 asks whether this pattern is present for other ${\mathcal{T}}_c^b$${\mathcal{T}}_c^b$T_(c)^(b)\mathscr{T}_{c}^{b}${\mathcal{T}}_c^b$, in other words for ${\mathcal{T}}_c^b\subset \mathcal{T}$${\mathcal{T}}_c^b⊂\mathcal{T}$T_(c)^(b)subT\mathscr{T}_{c}^{b} \subset \mathscr{T}${\mathcal{T}}_c^b⊂\mathcal{T}$ where $\mathcal{T}$$\mathcal{T}$T\mathscr{T}$\mathcal{T}$ are some other $R$$R$RR$R$-linear, approximable triangulated categories.

And finally:

Problem 7.12. Is there a version of Theorem 6.6 that holds for non-noetherian rings?

There is evidence that something might be true, see Ben-Zvi, Nadler, and Preygel [5, SECTION 3]. But the author has no idea what the right statement ought to be, let alone how to go about proving it.

We would like to thank Asilata Bapat, Jim Borger, Anand Deopurkar, Jack Hall, Bernhard Keller, Tony Licata, and Bregje Pauwels for corrections and improvements to earlier drafts. Needless to say, the flaws that remain are entirely the author's fault.

This work was partially supported by the Australian Research Council, more specifically by grants number DP150102313 and DP200102537.

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[4] K. Aoki, Quasiexcellence implies strong generation. J. Reine Angew. Math. (published online 14 August 2021, 6 pages), see also 2020, arXiv:2009.02013.

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